Integrand size = 21, antiderivative size = 140 \[ \int \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx=-\frac {61 a^4 x}{8}+\frac {2 a^4 \text {arctanh}(\cos (c+d x))}{d}+\frac {4 a^4 \cos ^3(c+d x)}{3 d}-\frac {5 a^4 \cot (c+d x)}{d}-\frac {a^4 \cot ^3(c+d x)}{3 d}-\frac {2 a^4 \cot (c+d x) \csc (c+d x)}{d}-\frac {19 a^4 \cos (c+d x) \sin (c+d x)}{8 d}-\frac {a^4 \cos (c+d x) \sin ^3(c+d x)}{4 d} \]
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Time = 0.19 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {2788, 3855, 3852, 8, 3853, 2718, 2715, 2713} \[ \int \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {2 a^4 \text {arctanh}(\cos (c+d x))}{d}+\frac {4 a^4 \cos ^3(c+d x)}{3 d}-\frac {a^4 \cot ^3(c+d x)}{3 d}-\frac {5 a^4 \cot (c+d x)}{d}-\frac {a^4 \sin ^3(c+d x) \cos (c+d x)}{4 d}-\frac {19 a^4 \sin (c+d x) \cos (c+d x)}{8 d}-\frac {2 a^4 \cot (c+d x) \csc (c+d x)}{d}-\frac {61 a^4 x}{8} \]
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Rule 8
Rule 2713
Rule 2715
Rule 2718
Rule 2788
Rule 3852
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {\int \left (-10 a^8-4 a^8 \csc (c+d x)+4 a^8 \csc ^2(c+d x)+4 a^8 \csc ^3(c+d x)+a^8 \csc ^4(c+d x)-4 a^8 \sin (c+d x)+4 a^8 \sin ^2(c+d x)+4 a^8 \sin ^3(c+d x)+a^8 \sin ^4(c+d x)\right ) \, dx}{a^4} \\ & = -10 a^4 x+a^4 \int \csc ^4(c+d x) \, dx+a^4 \int \sin ^4(c+d x) \, dx-\left (4 a^4\right ) \int \csc (c+d x) \, dx+\left (4 a^4\right ) \int \csc ^2(c+d x) \, dx+\left (4 a^4\right ) \int \csc ^3(c+d x) \, dx-\left (4 a^4\right ) \int \sin (c+d x) \, dx+\left (4 a^4\right ) \int \sin ^2(c+d x) \, dx+\left (4 a^4\right ) \int \sin ^3(c+d x) \, dx \\ & = -10 a^4 x+\frac {4 a^4 \text {arctanh}(\cos (c+d x))}{d}+\frac {4 a^4 \cos (c+d x)}{d}-\frac {2 a^4 \cot (c+d x) \csc (c+d x)}{d}-\frac {2 a^4 \cos (c+d x) \sin (c+d x)}{d}-\frac {a^4 \cos (c+d x) \sin ^3(c+d x)}{4 d}+\frac {1}{4} \left (3 a^4\right ) \int \sin ^2(c+d x) \, dx+\left (2 a^4\right ) \int 1 \, dx+\left (2 a^4\right ) \int \csc (c+d x) \, dx-\frac {a^4 \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{d}-\frac {\left (4 a^4\right ) \text {Subst}(\int 1 \, dx,x,\cot (c+d x))}{d}-\frac {\left (4 a^4\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d} \\ & = -8 a^4 x+\frac {2 a^4 \text {arctanh}(\cos (c+d x))}{d}+\frac {4 a^4 \cos ^3(c+d x)}{3 d}-\frac {5 a^4 \cot (c+d x)}{d}-\frac {a^4 \cot ^3(c+d x)}{3 d}-\frac {2 a^4 \cot (c+d x) \csc (c+d x)}{d}-\frac {19 a^4 \cos (c+d x) \sin (c+d x)}{8 d}-\frac {a^4 \cos (c+d x) \sin ^3(c+d x)}{4 d}+\frac {1}{8} \left (3 a^4\right ) \int 1 \, dx \\ & = -\frac {61 a^4 x}{8}+\frac {2 a^4 \text {arctanh}(\cos (c+d x))}{d}+\frac {4 a^4 \cos ^3(c+d x)}{3 d}-\frac {5 a^4 \cot (c+d x)}{d}-\frac {a^4 \cot ^3(c+d x)}{3 d}-\frac {2 a^4 \cot (c+d x) \csc (c+d x)}{d}-\frac {19 a^4 \cos (c+d x) \sin (c+d x)}{8 d}-\frac {a^4 \cos (c+d x) \sin ^3(c+d x)}{4 d} \\ \end{align*}
Time = 4.76 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.49 \[ \int \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {a^4 (1+\sin (c+d x))^4 \left (-732 (c+d x)+96 \cos (c+d x)+32 \cos (3 (c+d x))-224 \cot \left (\frac {1}{2} (c+d x)\right )-48 \csc ^2\left (\frac {1}{2} (c+d x)\right )+192 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-192 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+48 \sec ^2\left (\frac {1}{2} (c+d x)\right )+32 \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )-2 \csc ^4\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x)-120 \sin (2 (c+d x))+3 \sin (4 (c+d x))+224 \tan \left (\frac {1}{2} (c+d x)\right )\right )}{96 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^8} \]
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Time = 0.46 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.19
method | result | size |
parallelrisch | \(\frac {a^{4} \left (\csc ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\sec ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (384 \left (-3 \sin \left (d x +c \right )+\sin \left (3 d x +3 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1464 d x \sin \left (3 d x +3 c \right )-4392 d x \sin \left (d x +c \right )-672 \sin \left (2 d x +2 c \right )+128 \sin \left (3 d x +3 c \right )-32 \sin \left (6 d x +6 c \right )-1395 \cos \left (d x +c \right )+1265 \cos \left (3 d x +3 c \right )-129 \cos \left (5 d x +5 c \right )+3 \cos \left (7 d x +7 c \right )-384 \sin \left (d x +c \right )\right )}{6144 d}\) | \(167\) |
derivativedivides | \(\frac {a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+4 a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{3}+\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+6 a^{4} \left (-\frac {\cos ^{5}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )+4 a^{4} \left (-\frac {\cos ^{5}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{2}-\frac {3 \cos \left (d x +c \right )}{2}-\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+a^{4} \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )}{d}\) | \(222\) |
default | \(\frac {a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+4 a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{3}+\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+6 a^{4} \left (-\frac {\cos ^{5}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )+4 a^{4} \left (-\frac {\cos ^{5}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{2}-\frac {3 \cos \left (d x +c \right )}{2}-\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+a^{4} \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )}{d}\) | \(222\) |
risch | \(-\frac {61 a^{4} x}{8}-\frac {i a^{4} {\mathrm e}^{4 i \left (d x +c \right )}}{64 d}+\frac {5 i a^{4} {\mathrm e}^{2 i \left (d x +c \right )}}{8 d}+\frac {a^{4} {\mathrm e}^{i \left (d x +c \right )}}{2 d}+\frac {a^{4} {\mathrm e}^{-i \left (d x +c \right )}}{2 d}-\frac {5 i a^{4} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}+\frac {i a^{4} {\mathrm e}^{-4 i \left (d x +c \right )}}{64 d}+\frac {4 a^{4} \left (-6 i {\mathrm e}^{4 i \left (d x +c \right )}+3 \,{\mathrm e}^{5 i \left (d x +c \right )}+15 i {\mathrm e}^{2 i \left (d x +c \right )}-7 i-3 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}+\frac {2 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d}-\frac {2 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d}+\frac {a^{4} \cos \left (3 d x +3 c \right )}{3 d}\) | \(241\) |
norman | \(\frac {-\frac {a^{4}}{24 d}-\frac {a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d}-\frac {61 a^{4} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}-\frac {97 a^{4} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {93 a^{4} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {93 a^{4} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {97 a^{4} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {61 a^{4} \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}+\frac {a^{4} \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}+\frac {a^{4} \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}-\frac {61 a^{4} x \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}-\frac {61 a^{4} x \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {183 a^{4} x \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {61 a^{4} x \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {61 a^{4} x \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}-\frac {4 a^{4} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {4 a^{4} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {5 a^{4} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {47 a^{4} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {2 a^{4} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}\) | \(386\) |
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Time = 0.29 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.56 \[ \int \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx=-\frac {6 \, a^{4} \cos \left (d x + c\right )^{7} - 75 \, a^{4} \cos \left (d x + c\right )^{5} + 244 \, a^{4} \cos \left (d x + c\right )^{3} - 183 \, a^{4} \cos \left (d x + c\right ) - 24 \, {\left (a^{4} \cos \left (d x + c\right )^{2} - a^{4}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 24 \, {\left (a^{4} \cos \left (d x + c\right )^{2} - a^{4}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - {\left (32 \, a^{4} \cos \left (d x + c\right )^{5} - 183 \, a^{4} d x \cos \left (d x + c\right )^{2} - 32 \, a^{4} \cos \left (d x + c\right )^{3} + 183 \, a^{4} d x + 48 \, a^{4} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]
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Timed out. \[ \int \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx=\text {Timed out} \]
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Time = 0.52 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.56 \[ \int \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {64 \, {\left (2 \, \cos \left (d x + c\right )^{3} + 6 \, \cos \left (d x + c\right ) - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a^{4} + 3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{4} - 288 \, {\left (3 \, d x + 3 \, c + \frac {3 \, \tan \left (d x + c\right )^{2} + 2}{\tan \left (d x + c\right )^{3} + \tan \left (d x + c\right )}\right )} a^{4} + 32 \, {\left (3 \, d x + 3 \, c + \frac {3 \, \tan \left (d x + c\right )^{2} - 1}{\tan \left (d x + c\right )^{3}}\right )} a^{4} + 96 \, a^{4} {\left (\frac {2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} - 4 \, \cos \left (d x + c\right ) + 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{96 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 274 vs. \(2 (130) = 260\).
Time = 0.94 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.96 \[ \int \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 183 \, {\left (d x + c\right )} a^{4} - 48 \, a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 57 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {88 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 57 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{4}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}} + \frac {2 \, {\left (57 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 96 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 81 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 96 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 81 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 32 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 57 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 32 \, a^{4}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4}}}{24 \, d} \]
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Time = 9.57 (sec) , antiderivative size = 384, normalized size of antiderivative = 2.74 \[ \int \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2\,d}+\frac {a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,d}-\frac {2\,a^4\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {61\,a^4\,\mathrm {atan}\left (\frac {3721\,a^8}{16\,\left (61\,a^8-\frac {3721\,a^8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}\right )}+\frac {61\,a^8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{61\,a^8-\frac {3721\,a^8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}}\right )}{4\,d}+\frac {19\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d}-\frac {-19\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-60\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\frac {67\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{3}-48\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\frac {508\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}+\frac {8\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{3}+116\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-\frac {16\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}+\frac {61\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}+4\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {a^4}{3}}{d\,\left (8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+48\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\right )} \]
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